If we have a matrixA

and we construct a matrixM

, whose columns are just eigenvectors ofA

, then we always have

AM=MD

for some diagonal matrixD

. However, if we wish to use this matrix to write down a nice factored form forA

, we must take care when constructingM

to ensure thatM

is an invertible matrix.

Consider the matrixA=

(-18

-20

15

17

)

. This has eigenvectorsv

1

=(3

4

)

andv

2

=(1

1

)

,where

Av

1

=2v

1

andAv

2

=3v

2

.

lfwe have a matrix A and we construct a matrix M, whose columns are just eigenvectors olA , then we always have u AM 2 MD for some diagonal matrix D. However, if we wish to use this matrix to write down a nice factored form for A' we must take care when constructing M to ensure that M is an Invertible matrix. -18 15 Considerthe matrix A. = -20 17 ), This has eigenvecldrs v1 = (j) and v2 = ( _1) , where Avl : 2v1 and sz : 73"2. L: 3 3t) 4 40 (Le. dene L to be the matrix whose columns are v1 and 10v1), then we can calculate AL: 13 15 3 30 :|''> in 2t] 17 4 40 E 3 30 2 0 4 40 t] 2 =1. 2 0 D 2 ) is not invertible! So we cannot use this to diagonalise A. lfwe dene 1 2 Note: the Maple notation torthe matrix (3 4) is,>. 330 Since the columns DfL are lineany dependent, the matrix L = (4 If we instead dene whose columns are the linearly independent eigenvectors above' then we get M: 718 15 3 71 =l. la 720 17 4 7 [E 3 1 2 0 4 1 0 3 M 2 0 U 3 Since the columns of M are linearly independent, the matrix M is invertible, Namely M has inverse M-1 = l in a